Concept development, floating bridge E39 Bjørnafjorden Appendix E – Enclosure 7

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Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 7

10205546‐08‐NOT‐098 Bridge closure due to wind

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00 24.05.2019 Final issue K. Aas-Jakobsen R. M. Larssen S. E. Jakobsen

REV. DATE DESCRIPTION PREPARED BY CHECKED BY APPROVED BY

MEMO

PROJECT Concept development, floating bridge

E39 Bjørnafjorden DOCUMENT CODE 10205546-08-NOT-098

CLIENT Statens vegvesen ACCESSIBILITY Restricted

SUBJECT Bridge closure due to wind PROJECT MANAGER Svein Erik Jakobsen

TO Statens vegvesen PREPARED BY Ketil Aas-Jakobsen

COPY TO RESPONSIBLE UNIT AMC

SUMMARY

Bridge closure due to wind is discussed for the different bridge alignments in light of the turbulence data in the Metocean Design basis. The discussion shows that bridge closure due to wind may increase due to the increased turbulence intensity from the southern sector. We recommend that this is addressed by an updated analysis. To do these analysis sectorial long term distribution data of the wind is needed (e.g. Weibull parameters).

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1 Bridge closure due to wind

The Metocean design basis provided in this project indicate that closure due to wind effects on vehicles should be re-evaluated. For certain wind direction the turbulence intensity is specified to 30% /2/ and this will reduce the wind speed for which closure of the bridge must be considered . N400 states that the structure should be calculated with traffic load up to a 3s gust speed of 35 m/s (N400-§5.4.3.3 /4/). This can be understood as if the gust speed is above 35 m/s, no vehicles is on the bridge, i.e. the bridge is closed. The 3s gust speed can be approximated by U3s = U10min+ 3.5*σu, where σu=Iu*U10min. Thus, by increasing the turbulence intensity, U10min is reduced.E.g. by setting U3s

to 35 m/s, the closure wind speed becomes:

A. Iu=11% => U10min = 25.3 m/s B. Iu=30% => U10min = 17.1 m/s.

The criteria in A above give wind speed similar to closure wind speed used for other bridges

worldwide, and similar to the criteria in the previous phase of the Bjørnafjorden crossing. As can be seen in B, if one applies the same method for 30% turbulence intensity, a significantly lower closure speed is found. A reduction of closure speed to this level will increase closure time significantly, and it is questionable if the up-time target for the bridge can be reached without taking measures.

The current Metocean design basis does not contain data about long term distribution, neither for sectorial or omnidirectional data, but by using the results from the calculations in the previous phase the following is found /5/:

 U10min is above 25m/s 16 hours yearly

 U10min is above 20m/s 69 hours yearly

Thus, a criteria of 17 m/s will give significantly more bridge closing due to wind than the current value of about 16 hours.

In this phase of the project the client has not asked specifically for up-time evaluation of the concepts, as it has not been deemed necessary. Calculations in previous phases has shown that the uptime fulfils the criteria of 99.5%, which corresponds to 43.8 hours of year closure /5/. The above calculation indicates that it will be challenging to reach the 99.5% target without detailed

assessment.

In the previous analysis wind in the sector +/- 30 degrees to perpendicular to the vehicles has been viewed as particular important when assessing the effects, as the effective wind attack angle on the vehicle has a significant perpendicular component when one takes into account the speed of the vehicle itself. The critical direction compared to a linearized alignment segment is shown to the right in Figure 1.

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Concept development, floating bridge E39 Bjørnafjorden Bridge closure due to wind

Figure 1 Left: Wind rose and area of increased turbulence intensity (Yellow), compared to K12 (Red), K13 (Black) and K14 (Blue).

Right: Critical wind direction compared to alignment (red).

The left part of Figure 1 shows the wind rose with superimposed alignments; K12 (red), K13 (black) and K14 (blue), and the wind direction zone for which elevated turbulence should be applied. Table 1 show a summary of these two figures combined. As can be seen from the table, closure time on Concept K12 and K13 may be affected by the elevated turbulence intensity from the southern sector.

Based on this evaluation we recommend that up-time is re-evaluated for the concepts based on the current Metocean Design Basis. For this sectorial long term distribution is needed, including Weibull parameters.

Table 1 Critical cross wind.

Concept Alignment angle compared to North at

point X in Figure 1 (approx.)

Wind gust direction relative to alignment

Within critical region marked red

in Figure 1

K12 + 35ô 115ô – 175ô Yes, partly.

Detail assessment necessary

K13 + 10ô 200ô – 140ô Close. Detail assessment

necessary

K14 South: - 5^o

Mid point: +20^o

155ô – 215ô (South) 130ô – 190ô (Mid point)

South: No Close. Detail assessment

necessary X

120 60

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2 References

/1/ SBJ-32-C4-SVV-90-BA-001 - Design Basis Bjørnafjorden.

/2/ SBJ-01-C4-SVV-01-BA-001 - Metocean Design Basis.

/3/ NS-EN 1991-1-4:2005+NA:2009. Eurocode 1: Action on structures. Part 1-4: General actions – Wind actions

/4/ N400 Bruprosjektering. Statens Vegvesen.

/5/ Vurdering og sammenlikning av brukonsepter for kryssing av Bjørnafjorden. Oppetid. Versjon 1.0.

25.04.2016.

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Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 8

10205546‐08‐NOT‐176

Aerodynamic stability K11

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00 24.05.2019 Final issue A. Larsen K. Aas-Jakobsen S. E. Jakobsen

MEMO

PROJECT Concept development, floating bridge E39

Bjørnafjorden DOCUMENT CODE 10205546-08-NOT-176

SUBJECT Aerodynamic stability of K11 PROJECT MANAGER Svein Erik Jakobsen

TO Ketil Aas-Jakobsen PREPARED BY Allan Larsen

COPY TO Petter Sundquist RESPONSIBLE UNIT AMC

SUMMARY

This memo summarises the wind stability of the bridge girders of the K11 alternative for Bjørnafjorden bridge. Based on Håndbok N400 the bridge should be aerodynamically stable for speeds up to 81.7m/s.

The memo concludes that the K11 bridge is aerodynamically stable at wind speeds up to and beyond the requirements set by Håndbok N400 Bruprosjektering.

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Concept development, floating bridge E39 Bjørnafjorden Aerodynamic stability of K11

1 Wind load class

Following N400 /1/, section 5.4.3, a bridge shall be considered wind load class III when the following criteria apply:

 Highest eigen period > 2 s

 Span length > 300 m

Modal analyses of the K11 alternative for Bjørnafjorden floating bridge yields the highest eigen period to be 126 s. Further the main span of the cable stayed bridge is 380 m.

The above class III criteria are thus seen to be fulfilled. Hence verification of the wind stability of the bridge structure shall include interactions between the dynamics of the structure and wind field as well as aerodynamic stiffness and damping effects. The verification thus includes assessment of vortex induced vibrations /1/ section 5.4.3.7 and check of aerodynamic instabilities /1/ section 5.4.3.8.

2 Critical wind speed for onset of aerodynamic instabilities

Following N400 /1/, section 5.4.3 the critical wind speed for onset of aerodynamic instabilities shall be higher than 1.6 times the 500 year return period, 10 min mean wind speed at bridge girder level:

𝑉_{𝑐𝑟𝑖𝑡}(𝑧) > 1.6 ∙ 𝑉_𝑚(𝑧, 𝑇 = 600 𝑠, 𝑅 = 500 𝑦𝑒𝑎𝑟) (1) Following the MetOcean Design Basis /2/ the 50 year return period, 10 min mean wind speed at 𝑧 = 10 m level is 𝑉_𝑚(𝑧 = 10 𝑚, 𝑇 = 600 𝑠, 𝑅 = 50 𝑦𝑒𝑎𝑟) = 𝑉_𝑚,𝑏 = 30.5 m/s.

Extrapolation to the level of the cable stayed bridge (𝑧 = 65 m) proceeds following (2):

𝑉_𝑚(𝑧) = 𝐶_{𝑝𝑟𝑜𝑏}∙ 𝑉_𝑚,𝑏∙ 𝑘_𝑇∙ 𝑙𝑛 (𝑧

𝑧₀) (2)

/2/ defines 𝑘_𝑇 = 0.17 and 𝑧₀ = 0.01 m for the Bjørnafjord site.

𝐶_{𝑝𝑟𝑜𝑏} is a coefficient that transforms 50 year return wind speeds to other return periods 𝑅 /3/:

𝐶_{𝑝𝑟𝑜𝑏}=

√

1 − 0.2 ∙ 𝑙𝑛 (−𝑙𝑛 (1 −1 𝑅)) 1 − 0.2 ∙ 𝑙𝑛 (−𝑙𝑛 (1 − 1

50))

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For 𝑅 = 500 (3) yields 𝐶_{𝑝𝑟𝑜𝑏} = 1.122. Taking 𝑧 = 65 m, (1), (2) yields 𝑉_{𝑐𝑟𝑖𝑡}(65) > 81.7 m/s.

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3 Bridge girder aerodynamic properties

The present evaluation of the aerodynamic stability of the K11 bridge alternative is based on discrete vortex computations of steady state wind load coefficients and Aerodynamic Derivatives (flutter coefficients) for the SS1-b cross section, Figure 3.1.

The steady state wind load coefficients obtained in /4/ are reproduced in Table 3.1.

Aerodynamic derivatives calculated for the non-dimensional wind speed range 2.5 < 𝑉 𝑓𝐵⁄ < 25 are shown in Figure 3.2

Figure 3.1 Discrete vortex panel model of the SS1-b cross section geometry.

Table 3.1 Steady state wind load coefficients for the SS1-b cross section from discrete vortex simulations.

𝑪_𝑫𝟎 [ - ] 𝑪_𝑳𝟎 [ - ] 𝒅𝑪_𝑳⁄𝒅𝜶 [𝟏 𝒓𝒂𝒅⁄ ] 𝑪_𝑴𝟎 [ - ] 𝒅𝑪_𝑴⁄𝒅𝜶 [𝟏 𝒓𝒂𝒅⁄ ]

0.57 -0.06 3.72 -0.013 0.93*

*𝑑𝐶_𝑀⁄𝑑𝛼 is estimated assuming the aerodynamic centre to be located at the upwind ¼ chord point The wind load coefficients in Table 3.1 above are normalized the conventional way by the dynamic head of the wind ½𝜌𝑉² and a characteristic dimension of the cross section. The section depth 𝐻 = 3.5 m in case of the along wind drag loading and the cross section width 𝐵 = 30 m in case of the lift and overturning moment.

Figure 3.2 Aerodynamic derivatives for the SS1-b cross section from discrete vortex simulations.

The aerodynamic derivatives calculated at 6 discrete values of (𝑉 𝑓𝐵⁄ )_𝑗, (𝑗 = 1 − 6) shown in Figure 3.2. The displays the expected behaviour for non-dimensional wind speeds in the range 2.5 <

𝑉 𝑓𝐵⁄ < 18 displaying a monotonic growth in a linear or parabolic fashion. For the highest non- dimensional wind speeds 18 < 𝑉 𝑓𝐵⁄ < 25 it is noted that the 𝐻_2..4^∗ and 𝐴_2..4^∗ aerodynamic derivatives display an unexpected non-monotonic behaviour which may influence stability calculations slightly.

0 5 10 15 20 25

40 -

20 -

0 20 40

60 H1*

H2*H3*

H4*

V/fB

Vertical

0 5 10 15 20 25

10 -

0 10

20 A1*

A2*A3*

A4*

V/fB

Torsion

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4 Vortex induced vibrations

Vortex shedding in the wake of box girders may result in limited amplitude oscillations of the bridge girder at wind speeds where rhythmic vortex shedding locks on to a vertical bending or torsion eigen mode.

Practical experience from suspension bridges with shallow trapezoidal box girders have shown that vortex induced oscillations are usually confined to vertical modes only and occur at low wind speeds typically less than 12 m/s and for weather conditions with low atmospheric conditions.

Vortex induced vibrations of suspension bridges (Osterøy bridge, Norway and Storebælt East bridge, Denmark) has proven to be linked to severe flow separation and associated rhythmic vortex shedding at the knuckle line between the horizontal bottom plate and the lower inclined downwind side panel. Wind tunnel research /5/ has demonstrated that severe flow separation and vortex shedding can be avoided if the angle between the horizontal bottom plate and the lower side panels can be kept at approximately 15 deg. The 15 deg principle was recently introduced for the design of the girder of the Hålogaland Bridge, Norway and has proven to be free of vortex induced vibrations in full scale as well as in wind tunnel tests.

The design of the cross section shape of the girders of Bjørnafjorden bridge incorporates the 15 deg. principle. Thus, vortex induced vibrations are not expected to be an issue for the present design.

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5 Verification of aerodynamic instabilities

N400 specifies that a wind load class III bridge shall be verified for four types of aerodynamic instabilities:

 Galloping

 Static divergence

 Classical flutter

 Torsion instability

Each type of aerodynamic instability will be discussed in separate sections below.

Galloping

Galloping is a cross wind vertical instability resulting in onset of vertical divergent oscillations above a certain threshold wind speed.

A necessary condition for galloping to occur is that the lift slope 𝑑𝐶_𝐿⁄𝑑𝛼 is negative. With reference to Table 3.1 𝑑𝐶_𝐿⁄𝑑𝛼 = 3.72 > 0, thus galloping will not occur for the K11 design for Bjørnafjorden bridge.

Static divergence

Static divergence is a buckling type instability of the bridge girder occurring at the wind speed where the wind induced external moment acting on the girder exceeds the structural capacity. An estimate of the wind speed for onset for divergence 𝑉_𝑑𝑖𝑣 is given as /1/:

𝑉_𝑑𝑖𝑣 = 2𝜋𝑓_𝛼√ 2𝐼_𝛼𝑒𝑞

𝜌𝐵⁴𝑑𝐶_𝑀⁄𝑑𝛼 (4)

Where:

𝑓_𝛼 is the eigenfrequency of the lowest torsion mode having modal mass 𝑀_𝛼. 𝐼_𝛼𝑒𝑞 = ^𝑀^𝛼

∫ 𝜑(𝑥)₀^𝐿 ²𝑑𝑥 is the corresponding equivalent mass moment of inertia.

𝜌 = 1.25 kg/m³ is air density 𝐵 = over-all girder width

The lowest torsion mode of the bridge is mode 25 displaying three half waves along the low level floating bridge, Figure 5.1, having an eigenfrequency 𝑓_𝛼 = 0.157 Hz and a modal mass 𝑀_𝛼 = 107.213∙10⁶ kg.

Figure 5.1 First torsion mode (mode 25) of Bjørnafjord K11 Alternative.

Inserting 𝑑𝐶_𝑀⁄𝑑𝛼 = 0.92 and the above structural properties in (4) a divergence wind speed 𝑉_𝑑𝑖𝑣 = 2627 m/s which is well above the requirement of 81.7 m/s.

0 1.033 10³ 2.066 10³ 3.099 10³ 4.132 10³ 5.165 10³

0.1 -

0.05 -

0 0.05 0.1

Length [m]

Modal displacement

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Classical flutter

Classical flutter involves as a minimum two modes of motion. A torsion mode and a vertical bending mode of similar mode shape but with a lower eigenfrequency. The critical wind speed for onset of classical flutter is reached when the wind loading on the bridge girder makes the bending and torsion frequencies equal thereby establishing a resonant exchange of energy between to two modes. This in turn leads to divergent coupled torsion bending oscillations of the bridge girder. In cases where more vertical modes exist below the torsion mode these vertical modes may couple to form a compound vertical mode shape which couples with the torsion mode at the onset of flutter.

Different methods exist for calculation of the flutter wind speed of a bridge deck. The present method outlined in section 6 is an expansion of the AMC method (Air material Command) which allows an arbitrary number of modes and degrees of freedom of a bridge deck to couple into flutter /6/. The input to the flutter calculation constitutes the modes assumed to couple into flutter, the corresponding modal masses and eigenfrequencies and aerodynamic derivatives particular to the bridge deck.

The present multi-mode flutter analysis of Bjørnafjorden K11 alternative assumes that the first torsion mode and 9 vertical bending modes may couple into flutter, Figure 5.3. Eigenfrequencies and modal masses with of the modes are listed in Table 5.1.

From the resulting flutter diagrams Figure 5.2 it is noted that all apparent damping levels remain negative for all wind speeds below 120 m/s. Hence the bridge is stable against classical flutter up to and beyond the N400 requirement of 81.7 m/s regardless of structural damping.

Table 5.1 Lowest vertical and torsion modal masses and eigenfrequencies. Bjørnafjord K11 alternative.

Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Modal mass [kg] 35723∙10³ 21048∙10³ 28676∙10³ 69297∙10³ 75426∙10³

Eigenfrequency [Hz] 0.1596 0.1538 0.1546 01548 0.1548

Modal mass 60315∙10³ 98445∙10³ 53534∙10³ 75942∙10³ 107213∙10³

Eigenfrequency [Hz] 0.1548 0.1548 0.1550 0.1550 0.1567

Figure 5.2 Apparent aerodynamic damping 𝑔 as function of normalized and absolute wind speed for Bjornafjorden K11 alternative.

0 5 10 15 20 25

0.8 -

0.6 -

0.4 -

0.2 -

0 0.2

V/fB

Apparent damping g

0 20 40 60 80 100 120

0.8 -

0.6 -

0.4 -

0.2 -

0 0.2

Wind speed [m/s]

Apparent damping g

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Figure 5.3 Flutter modes for Bjørnafjorden bridge K11 alternatives. Modes 14 – 22 (red) are vertical Mode 25 (blue) is torsion.

Mode 14

Mode 15

Mode 16

Mode 17

Mode 18

Mode 19

Mode 20

Mode 21

Mode 22

Mode 25

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Torsion instability

Torsion instability is a condition resulting in onset of torsional divergent oscillations above a certain threshold wind speed. Torsion instability is associated with the formation and travel of large coherent vortex structures across the bridge deck /6/. This type of instability is often associated with bluff plate girder bridge decks.

A necessary condition for the occurrence of torsion instability is that the 𝐴₂^∗ aerodynamic derivative change sign from negative at low wind speeds to positive at higher wind speeds.

From Figure 3.2 it is noted that 𝐴^∗₂ remains negative for all non-dimensional wind speeds up to at least 𝑉 𝑓𝐵⁄ = 25. It may thus be concluded that Bjørnafjorden K11 alternative will not encounter torsion instability at wind speeds below a wind speed of 25∙ 𝑓_𝛼𝐵 = 117 m/s which is well above the N400 requirement.

6 Multimode flutter theory

The calculation of the critical wind speed for onset of flutter follows from solving the complex valued eigenvalue problem (5) which combines the modal and aerodynamic properties of one torsion mode 𝛼(𝑥) and 𝑛 vertical bending modes ℎ1(𝑥) … ℎ𝑛(𝑥), /6/.

𝐷𝑒𝑡 [

1 + 𝐴_{𝛼𝛼,𝑗}− 𝜆_1,𝑗 ⋯ 𝐴_{𝛼ℎ𝑛,𝑗}

⋮ ⋱ ⋮

𝐻_{ℎ𝑛𝛼,𝑗}(𝑓_𝛼 𝑓_ℎ𝑛)

2

⋯ (1 + 𝐻_{ℎ𝑛ℎ𝑛,𝑗}) (𝑓_𝛼 𝑓_ℎ𝑛)

2

− 𝜆_𝑛,𝑗 ]

= 0 (5)

The individual elements in (5) combine modal and aerodynamic characteristics of the bridge deck girder and are composed as follows:

𝐴_{𝛼𝛼,𝑗} =𝜌𝐵⁴

𝑀_𝛼 𝐶_𝛼𝛼(𝐴_3,𝑗^∗ + 𝑖𝐴_2,𝑗^∗ ), 𝐶_𝛼𝛼= ∫ 𝛼(𝑥)²𝑑𝑥

𝐿

0

𝐴_{𝛼ℎ𝑛,𝑗} =𝜌𝐵⁴

𝑀_𝛼 𝐶_𝛼ℎ𝑛(𝐴_4,𝑗^∗ + 𝑖𝐴₁^∗, 𝑗), 𝐶_𝛼ℎ𝑛 = ∫ 𝛼(𝑠)ℎ𝑛(𝑥)𝑑𝑥

𝐿

0

𝐻_{ℎ𝑛𝛼,𝑗}=𝜌𝐵²

𝑀_ℎ𝑛𝐶_𝛼ℎ𝑛(𝐻_3,𝑗^∗ + 𝑖𝐻_2,𝑗^∗ ), 𝐶_ℎ𝑛𝛼 = ∫ 𝛼(𝑠)ℎ𝑛(𝑥)𝑑𝑥

𝐿

0

𝐻_{ℎ𝑛ℎ𝑛,𝑗} =𝜌𝐵²

𝑀_ℎ𝑛𝐶_{ℎ𝑛ℎ𝑛}(𝐻_4.𝑗^∗ + 𝑖𝐻_1.𝑗^∗ ), 𝐶_{ℎ𝑛ℎ𝑛} = ∫ ℎ𝑛(𝑥)²𝑑𝑥

𝐿

0

(6)

The unknown to be solved for is the flutter frequency 𝑓 which is embedded in the eigenvalues 𝜆_𝑛,𝑗 through the identity:

𝑅𝑒(𝜆_𝑛,𝑗) + 𝑖 𝐼𝑚(𝜆_𝑛,𝑗) = (1 + 𝑖𝑔_𝑛,𝑗) (𝑓_𝛼 𝑓)

2

(7) Where 𝑔_𝑛,𝑗 is the apparent aerodynamic damping (negative) of a given mode at a given non- dimensional wind speed 𝑉 𝑓𝐵⁄ _𝑗.

Once the complex eigenvalues 𝜆_𝑛,𝑗 are determined for each of the non-dimensional wind speeds 𝑉 𝑓𝐵⁄ _𝑗 = 𝑉_𝑗^∗ for which the flutter derivatives are available, the equivalent aerodynamic damping and corresponding wind speed are obtained as:

𝑔_𝑛,𝑗 =𝐼𝑚(𝜆_𝑛,𝑗)

𝑅𝑒(𝜆_𝑛,𝑗), 𝑉_𝑗= 𝑉_𝑗^∗ 𝑓_𝛼𝐵

√𝑅𝑒(𝜆𝑛,𝑗) (8)

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By plotting the equivalent aerodynamic damping 𝑔 as function of the wind speed the critical wind speed is identified where the apparent damping 𝑔_𝑛,𝑗 equals twice the structural damping:

𝑔_𝑛,𝑗 = 2𝜁 (9)

Example, critical wind speed of Storebælt East Bridge

The above procedure is illustrated in the example below which pairs the aerodynamic derivatives shown in Figure 3.2 with the structural properties of Storebælt East bridge section model (unity modes along the span) for which wind tunnel measurement of the critical wind speeds are reported in the literature /7/.

Storebælt East bridge section model structural data (two modes):

Mass / unit length: 𝑚 = 22.74∙10³ kg/m

Mass moment of inertia / unit length: 𝐼 = 2.47∙10⁶ kg/m Vertical bending frequency: 𝑓_ℎ = 0.1 Hz

Torsion frequency: 𝑓_𝛼 = 0.278 Hz Deck width: 𝐵 = 31 m

Structural damping: 𝜁 = 0.003

Determination of the critical wind speed for onset of flutter following the above method is shown in Figure 6.1. It is noted that the red branch remains negative for all wind speeds. The purple branch starts being negative at low wind speeds but intersects the blue horizontal line (twice the structural damping) at a non-dimensional wind speed at 𝑉 𝑓𝐵⁄ = 12.3 (left diagram) corresponding to a critical wind speed of about 79 m/s (right diagram) which may be compared to a critical wind speed in the range 70 – 74 m/s measured in the wind tunnel /7/.

Figure 6.1 Determination of the critical wind speed for onset of flutter in non-dimensional for (left) and actual wind speed (right). Example: Storebælt East Bridge dynamic properties paired with SS1-b aerodynamic dericatives.

The geometry of the Storebælt East bridge and the SS1-b cross section is not identical, thus a perfect match of the above flutter calculation and the Storebælt wind tunnel tests can not be expected. However, the relatively close match is quite satisfactory and supports the credibility of the computed aerodynamic derivatives.

0 5 10 15 20 25

0.8 -

0.6 -

0.4 -

0.2 -

0 0.2

V/fB

Apparent damping g

0 20 40 60 80 100

0.8 -

0.6 -

0.4 -

0.2 -

0 0.2

Wind speed [m/s]

Apparent damping g

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7 References

/1/ N400: Håndbok, Bruprojektering, Statens vegvesen

/2/ MetOcean Design Basis. Document nr.: SBJ-01-C4-SVV-01-BA-001, 14-11-2018 including addendum of 18-03-2019.

/3/ Eurocode 1: Action on structures – Part 1-4: General actions – Wind actions. 2ed, 2007.

/4/ Memo 10205546-08-NOT-062: CFD analysis of cross section.

/5/ Larsen, A., Wall, A.: Shaping of bridge box girders to avoid vortex shedding response.

Journal of Wind Engineering and Industrial Aerodynamics 104-106, (2012) 159-165.

/6/ Larsen, A.: Bridge Deck Flutter Analysis. Proceedings of the Danish Society for Structural Science and Engineering, Vol. 87, No. 2-4, 2016.

/7/ Larsen, A.: Aerodynamic aspects of the final design of the 1624 m suspension bridge across the Great Belt. Journal of Wind Engineering and Industrial Aerodynamics 48, (1993) 261- 285.

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Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 9

10205546-08-NOT-183

Inhomogeneity in wind – effects on K12

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0 15.08.2019 Final issue K. Aas-Jakobsen R. M. Larssen S. E. Jakobsen

INTERNAL - MEMO

SUBJECT Inhomogeneity in wind – effects on K12 PROJECT MANAGER Svein Erik Jakobsen

TO Statens vegvesen PREPARED BY Ketil Aas-Jakobsen

SUMMARY

This note shows results from calculation of dynamic transverse response due to wind for concept K12, for four different variations of wind speed along the alignment. Results of variations of the parameters found in Table 15 in the Metocean Design basis is also shown. For the different distributions of the wind speed along the alignment, the case with constant distribution gives the highest dynamic wind response for all directions. Dynamic wind response of xLu, Au, Cux and Cuy is calculated and compared to N400 values. K12 is most sensitive to change in Cuy when the wind is more or less perpendicular to the alignment. For wind in the sector along the alignment the response is less

sensitive to change of the calculated parameters.

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1 Introduction

Several floating bridge concepts are proposed for the Bjørnafjorden crossing. This document explore the effect of inhomogeneity in wind the K12 concept, which is the curved bridge with side anchors.

The main focus will be on inhomogeneity effects on dynamic response for K12. K12 is a curved in the horizontal plane and transverse force is carried by the its shape and additional side anchors.

The concept is illustrated in Figure 1.

The calculations are done with a simplified, but fast method capable of varying input parameters including wind from all angles and variation of mean wind along the alignment. Results are presented as standard deviations of transverse displacement and strong axis moment. The basic idea is to utilise the eigenmodes of the structure to calculate the response and sum the modal response over all relevant modes. This is a well proven method for calculating dynamic response from wind. However, the simplified method has several disadvantages; currently it has a simplified load model implemented, it does not currently handle geometric stiffness, and damping introduced through the wetted part of the structure is introduced as static modal values for each mode, calculated for the base case with wind perpendicular to the structure. A simplified model is used to adjust the damping for different direction of the wind.

Figure 1 K12 General arrangement.

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2 Implementation – calculation of displacement

Calculation of displacement

Calculation of dynamic response from wind is simplified by including only the resonant part. I.e. the background part is discarded. For lightly damped structures calculations in other projects have shown that this will underestimate the dynamic response with about 5-10%.

The formulation in Eq. 6.50 in /6/ is used for calculating modal response:

𝑄(𝑓_𝑒) = 𝜌 ∑ ∑ √(𝐶_𝐷𝐻𝑉)_𝑖 𝑆_𝑢(𝑥_𝑖, 𝑓)(𝐶_𝐷𝐻𝑉)_𝑗 𝑆_𝑢(𝑥_𝑗, 𝑓_𝑒)√𝑐𝑜ℎ(𝑓_𝑒, ∆𝑠)𝜑_𝑖𝜑_𝑗

𝐿

𝑗 𝐿

𝑖

(1) Where f is the eigenfrequency of the mode and CD, H, V, Su and ϕ is evaluated individually for every element, and √coh is based on the separation of the point and the associated mid-point value.

The standard deviation for the response of each modes is obtained by

𝜎_𝑦(𝑓_𝑒) = √ 𝜋𝑓_𝑒 4𝜁_𝑚𝑄(𝑓_𝑒) (2𝜋𝑓_𝑒)⁴𝑀_𝑚²

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Where fe, Mm, and ζm are modal values for the particular mode. For ζm the structural damping is given as input and the aerodynamic damping is calculated. Viscous and hydrodynamic modal damping from the pontoons and line damping are taken from a separate analysis where the wind is perpendicular to the alignment. In order to compensate for different damping levels of viscous and line damping for different angle of attack, an empirical formula for compensation was developed in another part of this project, based on comparison with Orcaflex results. The formulae and principle is shown in Figure 2.

Figure 2 Empirical adjustment of damping.

The overall response of the included modes is calculated by the Root-Square-Sum method.

Typically, 20 modes is included. Moment values are calculated by differentiation.

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Results are presented in a polar diagram similar to the one shown in Figure 3. The angle in this diagram reflects the direction the wind is coming from. 0 degrees is wind from north and 90 degrees is wind from East. A sketch of the alignment is shown as a broken black line.

Figure 3 Principles for response presentation.

Analysis input parameters - wind

The analysis was conducted for wind response only. It was selected to use wind speed with 100year return period and as a first step it was selected to calculate response with the same turbulence intensity for all directions. Thus, standard N400 formulas and values are used in these reference calculations and the wind input to the analysis is:

Basic wind speed at z=10m : vb=25.2m/s Surface roughness: z0=0.01

Length scale at z=10m: ^xLu=100m Spectral shape parameter: Au=6.8

Decay parameters: Cux=3, Cuy=10, Cuz=10,

Figure 4 Left: Mean wind profile. Right: Spectral distribution with first 20 modes indicated (K12).

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Analysis input parameters – structural

The analysis mode is imported from a master sheet which also is used to build the Orcaflex model.

Thus, the geometry is the same as used in the Orcaflex analysis. Modal properties are derived from a NovaFrame analysis with the same structural and geometric properties. This analysis also

estimates hydrodynamic, viscous, aerodynamic and line damping for each mode.

Structural damping is generally set to the target value of 0.5% of critical. Since Orcaflex implements damping as Rayleigh damping, it is also implemented in this way in these calculations. The Rayleigh damping for the Orcaflex model is set to the target values for a low and high frequency. Due to the nature of Rayleigh damping the structural damping used in the current calculation is lower than the target value i.e. conservative.

Model used in calculations in this report: K12-06

The simplified load model used in these calculations only take into account the drag loading (See equation (1)). For simplicity CD is extracted for zero angle of attack. The values used for the high and low bridge, Figure 5 and Figure 6, is given in Table 1. In addition drag contributions from columns and cable stays are added at the correct location. Thus, the average Cd of the system is 0.88.

Figure 5 High bridge girder

Figure 6 Low bridge girder

Table 1 Wind load parameters used in calculations.

H [m]

B [m]

CD dCD

[1/rad]

CL dCL

[1/rad]

CM dCM

[1/rad]

High bridge 3.5 30.2 0.67 0.0 0.0 0.0 0.0 0.0

Low Bridge 4.0 30.2 0.80 0.0 0.0 0.0 0.0 0.0

B H

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Modes

No of modes used in calculations: 20 (T1=56.8s – T20=6.48s) is shown in Figure 7. Each plot shows transverse modal displacement as red, vertical modal displacement as green and torsional modal rotation as blue. Torsional contribution is scaled up by a factor of 10 to be visible on the plots as can be seen the transverse response dominates the first 14 modes.

Mode 1 – 10 Mode 11-20

Figure 7 Modeplot of first 20 modes of K12_06. Red: Transverse direction, Green: Vertical direction. Blue: torsional component scaled by a factor of 10.

Load model

Currently, only a simplified load in the drag direction is implemented in this method, which only takes into account the fluctuating u-component of the wind. Skew wind is include by a

decomposition of the force as shown below. When developing the method further more advanced load models can be implemented. The following load model is used:

FD = 1/2*ρ*(CD* cos(α))*H*U²

3 Verification of calculation method.

A flexible python script is developed based on the method described in Section 2.1 and checked versus hand-calculations (Mathcad) and full time-domain simulations in Orcaflex before parameter variations are reported.

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Verification versus hand-calculations

The method described in Section 2.1 is implemented in a Mathcad sheet, see Appendix A. Modal parameters and damping values are imported from Excel sheets. These sheets contain the same modal values as the ones used in the Orcaflex analysis.

The Mathcad sheet only takes into account wind on the girder. Since the Python script also takes into account wind area on cables and columns, an average Cd values is calculated from the Python input and used in Matchad. The weighted average value of the drag coefficients is 0.88 in Python, thus this value is used in the verification.

Mode-by-mode transverse displacement is calculated both in Mathcad and Python and compared for the case with mean wind perpendicular to the main alignment axis. Result comparison of standard deviation of transverse displacement is give in Table 2. As can be seen, the comparison is good both mode-by-mode and RSS over the 5 modes.

Table 2 Comparison of Mathcad and Python script for K12_06. Transverse displacement.

Mode Frequency

[Hz]

Total damping [% of crit]

Mathcad – std [m]

Python – std [m]

1 0.018 3.400% 0.930 0.927

2 0.023 3.955% 0.517 0.533

3 0.032 1.988% 0.322 0.315

4 0.046 1.593% 0.145 0.138

5 0.117 4.167% 0.107 0.096

RSS 1.126 1.127

Figure 8 shows a comparison along the alignment between Mathcad and Python, which shows that the agreement is good.

Figure 8 Comparison along alignment. Top: Matchad. Bottom: Python script.

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Verification versus Orcaflex

Verification of the simplified method was done my matching results with Orcaflex for the same input values. Orcaflex calculates the response in the time domain, and are thus, able to include non-linear effects of loading and response. Since Orcaflex has been verified and benchmarked with other calculation codes its reference for this verification.

The simplified method is based on a frequency domain approach, which is significantly faster than the time domain approach used by Orcaflex, but is not capable of handling nonlinearities in e.g.

hydrodynamics and aerodynamics, thus, exact match of results are not expected. Results of the comparison are shown in Figure 9. The left figure shows the standard deviation of the

displacement, while the right figure shows the calculated strong axis moment. Red dots are Orcaflex values (average of 10 runs), while the blue thick line is results from the simplified method for all wind angles and the doted lines indicate +/- 10% of the simplified method.

Figure 9 Comparison of standard deviation between simplified method and Orcaflex.

Blue solid line: Simplified method

Blue dotted lines: Simplified method +/- 10%.

Red dots: Orcaflex (average of 10 runs).

Black broken line: Alignment of K12.

The main purpose of this note is to study the effects and trends of varying wind parameters. Thus, the most important part of the comparison is to see that the simplified model is able to follow the major trends of the Orcaflex calculations.

Displacement results

For wind from east that are close to perpendicular to the alignment good agreement is found on comparison of displacement, while the results deviate more for wind from north and south. For wind from west the displacement shows good agreement. There are some differences in the physics that the models are able to represent that may contribute to the differences.

Possible sources for the deviations are:

- The simplified Python script uses the eigenfrequency only, and thus, discards the background value. Orcaflex uses all load frequencies in response calculations and thus includes the background response. Inclusion of background response will increase the response compared to using only the resonant part. A simplified analysis estimates the background response to about 6% on the standard deviation for the first mode of this structure, see the appendix for calculation.

Mathcad hand calculation

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- Effects of geometric stiffness that slightly changes the vibration frequencies, which again changes how the structure attracts load from the surroundings. This geometric effect is automatically included in Orcaflex, while not included in the modal input in the simplified frequency domain analysis. This is consistent with wind from east increasing the

compression in the arc, thus reducing the stiffness, which leads to reduced frequency, and thus, increased response. A simplifed sensitivity study on the above example gave a 7%

increase of response for a 3% reduction of vibration frequencies (i.e. softening due to compression in the members).

Strong axis moment

For the strong axis moment the simplified model follows the trend of Orcaflex for all checked wind attack angles. It is noted that larger deviations are found here than for displacement and that the simplified model underestimate the strong axis moment. This is most likely due to a more coarse segmentation of the simplified model, thus not resolving the curvature with the precision needed to get accurate results.

Even though the accuracy of this simplified method is not good enough for design, it is considered good enough for the purpose of this analysis; to look at the effect and differences in response for varying wind.

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4 Analysis and Results

Introduction

The following pages shows maximum transverse displacement as standard deviation for all wind directions for each wind scale types (Root Square Sum values). Results are plotted against the case with uniform wind along the alignment and with standard N400 values. Only wind from the easterly direction is calculated, as the verification calculations (above) showed a fairly symmetric response pattern along the mean alignment.

The following legends is used:

Blue solid thick line: Reference value (Results with N400 values) Blue dotted lines: Reference value +/- 10%

Black broken line : alignment of the concept.

Colored lines: result from parameter variation.

Mean wind variation along the alignment

Figure 10 describes and shows the variation of mean wind along the alignment. The left part of the figure represent the mean wind in the south end of the structure, while the right part of the figure represent the wind in the north end of the structure.

wScaleType Shape

wScaleType=1 wMin=1;wMax=1 Constant=1.0

wScaleType=2

wMin=0.6;wMax=1.0 Linear 0.6->1.0

wScaleType=3;

wMin=1.0;wMax=0.6 Linear 1.00->0.6

wScaleType=4;

wMin=0.8;wMax=1.0 Bi-linear (mid peak):

0.8->1.0->0.8

Figure 10 Wind variation along the alignment.

A summary of the results is shown in Figure 11. As can be seen the general trend is that constant mean wind (Wind Scale Type 1) gives the highest dynamic response regardless of direction. Second largest effect is from Wind Scale Type 4. Both these cases is symmetric about the mean. The two

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remaining Wind Scale Types are un-symmetric, and the maximum of the two is dependent on the direction of the wind in relation to the structure.

Figure 11 Dynamic response from varying mean wind profile along the deck, see Figure 10. The variation is from south towards north. I.e. for the yellow line the mean wind varies from 0.6 in the south to 1.0 in the north.

Variation in Table 15

4.3.1 Introduction

In Metocean Design basis /2/ Table 15 indicate that parameters in the wind varies. The key parameters are presented as P-values, representing probability of occurrence lower than the specified value. In orderer to understand the effect og the variation several relevant parameters were calculated and compared to the reference value; uniform wind as documented in Section 4.2.

Table 3 gives the variations used.

Table 3 Extract from "Table 15". Input varied below marked with blue.*Cux not given in N400. ESDU value is used in reference calculations.

Metocean Design Basis Analysis input (z=10m) (scaled based on N400)

N400 P10 P50 P90 P10 P50 P90

z [m] 10,0 50,0 50,0 50,0 10,0 10,0 10,0

xLu [m] 100,0 108,0 232,0 586,0 66,6 143,2 361,6

xLv [m] 25,0 50,0 141,0 472,0 30,9 87,0 291,2

xLw [m] 8,3 21,0 40,0 81,0 13,0 24,7 50,0

Au 6,8 3,80 7,30 16,30 3,8 7,3 16,3

Av 9,4 5,60 13,30 32,50 5,6 13,3 32,5

Aw 9,4 7,70 12,30 18,20 7,7 12,3 18,2

Cux* 3,0 - - - 5,0 7,0 10,0

Cuy 10,0 6,40 9,00 10,8 6,40 9,00 10,8

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4.3.2 Variation of xLu

For variation of xLu the response follows the reference value within +/- 10%. Low and high xLu values give lowest response. The results indicate that the highest response value for this concept is for an xLu value in medium range.

Figure 12 xLu variation. Left: std of displacement. Right: std of strong axis moment.

4.3.3 Variation of Au

For variation of Au the response follows the reference value within +/- 10%. The general trend is that increased Au values gives increase response.

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4.3.4 Variation of Cux

For variation of Cux the response follows the reference value within +/- 10%. The effect of change of cux is small for wind perpendicular to the alignment and more pronounced as the wind comes along the alignment. The Cux value of 3.0 used in the reference case gives the highest value of the calculated cases and increase of Cux value give reduced response.

4.3.5 Variation of Cuy

The calculation of variations of Cuy is shown in Figure 15. As can be seen the response is sensitive to change of Cuy, particularly for wind perpendicular to the alignment. Lower value of Cuy gives higher response and Cuy values below the P10 value of 6.4 increases the response with more than 20%. For skew wind, where the wind is more along the alignment, the effect is less pronounced.

Figure 15 Cuy variation. Left: std of displacement. Right: std of strong axis moment.

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5 Results and Discussion

Wind response of K12 for N400 standard wind values is calculated with a simplified method. The model is extracted from Orcaflex and contain the same modal parameters and damping, including hydrodynamic effects. The simplified method is compared to hand calculations and Orcaflex and shows fairly good agreement.

For the different distribution of the wind speed along the alignment, the case with constant distribution gives the highest dynamic wind response for all directions.

Sensitivity within the P10-P90 values is compared to the values given in N400. Of the studied parameters, xLu, Au, Cux and Cuy, Cuy gives the largest difference compare to N400.

For wind more or less perpendicular to the alignment Cuy gives the largest changes compared to N400 values. For wind along the alignment the sensitive to change is larges for Cux, but small in general.

6 References

/1/ SBJ-32-C4-SVV-90-BA-001 - Design Basis Bjørnafjorden.

/2/ SBJ-01-C4-SVV-01-BA-001 - Metocean Design Basis.

/3/ NS-EN 1991-1-4:2005+NA:2009. Eurocode 1: Action on structures. Part 1-4: General actions – Wind actions

/4/ N400 Bruprosjektering. Statens Vegvesen.

/5/ Vurdering og sammenlikning av brukonsepter for kryssing av Bjørnafjorden. Oppetid.

Versjon 1.0. 25.04.2016.

/6/ Theory of Bridge Aerodynamics. E.Strømmen. Springer. 2006.

7 Appendices

Mathcad sheet – Simplified dynamic wind calculation model Mathcad Sheet – Full dynamic wind calculation model.

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Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 10

10205546-08-NOT-184

Aerodynamic stability of K12

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0 15.08.2019 Final issue A. Larsen R. M. Larssen S. E. Jakobsen

MEMO

SUBJECT Aerodynamic stability of K12 PROJECT MANAGER Svein Erik Jakobsen

TO Statens vegvesen PREPARED BY Allan Larsen

SUMMARY

This memo summarises the wind stability of the bridge girders of the K12 alternative for Bjørnafjorden bridge.

The memo concludes that the bridge is aerodynamically stable at wind speeds up to and beyond the requirements set by Håndbok N400 Bruprosjektering.

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1 Wind load class ... 3 2 Critical wind speed for onset of aerodynamic instabilities ... 3 3 Bridge girder aerodynamic properties ... 3 4 Vortex induced vibrations ... 4 5 Verification of aerodynamic instabilities ... 5 Galloping ... 5 Static divergence ... 5 Classical flutter ... 6 Torsion instability ... 8 6 Multimode flutter calculation ... 9 Flutter calculation, Storebælt East bridge ... 10 7 References ... 11

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1 Wind load class

Following N400 /1/, section 5.4.3, a bridge shall be considered wind load class III when the following criteria apply:

 Highest eigen period > 2 s

 Span length > 300 m

Modal analyses of the K12 alternative for Bjørnafjorden floating bridge yields the highest eigen period to be 56.8 s. Further the main span of the cable stayed bridge is 380 m.

The above class III criteria are thus seen to be fulfilled. Hence verification of the wind stability of the bridge structure shall include interactions between the dynamics of the structure and wind field as well as aerodynamic stiffness and damping effects. The verification thus includes assessment of vortex induced vibrations /1/ section 5.4.3.7 and check of aerodynamic instabilities /1/ section 5.4.3.8.

2 Critical wind speed for onset of aerodynamic instabilities

Following N400 /1/, section 5.4.3 the critical wind speed for onset of aerodynamic instabilities shall be higher than 1.6 times the 500 year return period, 10 min mean wind speed at bridge girder level:

𝑉_{𝑐𝑟𝑖𝑡}(𝑧) > 1.6 ∙ 𝑉_𝑚(𝑧, 𝑇 = 600 𝑠, 𝑅 = 500 𝑦𝑒𝑎𝑟) (1) Following the MetOcean Design Basis /2/ the 50 year return period, 10 min mean wind speed at 𝑧 = 10 m level is 𝑉_𝑚(𝑧 = 10 𝑚, 𝑇 = 600 𝑠, 𝑅 = 50 𝑦𝑒𝑎𝑟) = 𝑉_𝑚,𝑏 = 30.5 m/s.

Extrapolation to the level of the cable stayed bridge (𝑧 = 65 m) proceeds following (2):

𝑉_𝑚(𝑧) = 𝐶_{𝑝𝑟𝑜𝑏}∙ 𝑉_𝑚,𝑏∙ 𝑘_𝑇∙ 𝑙𝑛 (𝑧

𝑧₀) (2)

/2/ defines 𝑘_𝑇 = 0.17 and 𝑧₀ = 0.01 m for the Bjørnafjord site.

𝐶_{𝑝𝑟𝑜𝑏} is a coefficient that transform 50 year return wind speeds to other return periods 𝑅 /3/:

𝐶_{𝑝𝑟𝑜𝑏}=

√

1 − 0.2 ∙ 𝑙𝑛 (−𝑙𝑛 (1 −1 𝑅)) 1 − 0.2 ∙ 𝑙𝑛 (−𝑙𝑛 (1 − 1

50))

(3)

For 𝑅 = 500 (3) yields 𝐶_{𝑝𝑟𝑜𝑏} = 1.122. Taking 𝑧 = 65 m, (1), (2) yields 𝑉_{𝑐𝑟𝑖𝑡}(65) > 81.7 m/s.

3 Bridge girder aerodynamic properties

The present evaluation of the aerodynamic stability of the K12 bridge alternative is based on discrete vortex computations of steady state wind load coefficients and Aerodynamic Derivatives (flutter coefficients) for the K12 cross section, Figure 3.1.

The steady state wind load coefficients obtained in /4/ are reproduced in Table 3.1.

Aerodynamic derivatives calculated for the non-dimensional wind speed range 4 < 𝑉 𝑓𝐵⁄ < 30 are shown in Figure 3.2

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Figure 3.1 Discrete vortex panel model of the K12 cross section geometry.

Table 3.1 Steady state wind load coefficients for the K12 cross section from discrete vortex simulations.

𝐶_𝐷0 [ - ] 𝐶_𝐿0 [ - ] 𝑑𝐶_𝐿⁄𝑑𝛼 [1 𝑟𝑎𝑑⁄ ] 𝐶_𝑀0 [ - ] 𝑑𝐶_𝑀⁄𝑑𝛼 [1 𝑟𝑎𝑑⁄ ]

0.675 -0.413 3.122 0.012 0.916

The wind load coefficients in Table 3.1 above are normalized the conventional way by the dynamic head of the wind ½𝜌𝑉² and a characteristic dimension of the cross section. The section depth 𝐻 = 4.0 m in case of the along wind drag loading and the cross section width 𝐵 = 30.2 m in case of the lift and overturning moment.

Figure 3.2 Aerodynamic derivatives for the K12 cross section from discrete vortex simulations.

The simulated aerodynamic derivatives shown in Figure 3.2 display the expected behaviour for non- dimensional wind speeds in the range 4 < 𝑉 𝑓𝐵⁄ < 18 displaying a monotonic growth in a linear or parabolic fashion. For the highest non-dimensional wind speeds 18 < 𝑉 𝑓𝐵⁄ < 30 it is noted that the 𝐻_2.4^∗ and 𝐴_2.4^∗ aerodynamic derivatives displays an unexpected non-monotonic behaviour which may influence stability calculations slightly.

4 Vortex induced vibrations

Vortex shedding in the wake of box girders may result in limited amplitude oscillations of the bridge girder at wind speeds where rhythmic vortex shedding locks on to a vertical bending or torsion eigen mode.

0 6 12 18 24 30

40 -

20 -

0 20 40

60 H1*

H2*H3*

H4*

V/fB

Vertical

0 6 12 18 24 30

10 -

0 10

20 A1*

A2*A3*

A4*

V/fB

Torsion

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Practical experience from suspension bridges with shallow trapezoidal box girders have shown that vortex induced oscillations are usually confined to vertical modes only and occur at low wind speeds typically less than 12 m/s and for weather conditions with low atmospheric conditions.

Vortex induced vibrations of suspension bridges (Osterøy bridge, Norway and Storebælt East bridge, Denmark) has proven to be linked to severe flow separation and associated rhythmic vortex shedding at the knuckle line between the horizontal bottom plate and the lower inclined downwind side panel. Wind tunnel research /5/ has demonstrated that severe flow separation and vortex shedding can be avoided if the angle between the horizontal bottom plate and the lower side panels can be kept at approximately 15 deg. The 15 deg principle was recently introduced for the design of the girder of the Hålogaland Bridge, Norway and has proven to be free of vortex induced vibrations in full scale as well as in wind tunnel tests.

The design of the cross section shape of the girders of Bjørnafjorden bridge incorporates the 15 deg. principle. Thus, vortex induced vibrations are not expected to be an issue for the present design.

5 Verification of aerodynamic instabilities

N400 specifies that a wind load class III bridge shall be verified for four types of aerodynamic instabilities:

 Galloping

 Static divergence

 Classical flutter

 Torsion instability

Each type of aerodynamic instability will be discussed in separate sections below.

Galloping

Galloping is a cross wind vertical instability resulting in onset of vertical divergent oscillations above a certain threshold wind speed.

A necessary condition for galloping to occur is that the lift slope 𝑑𝐶_𝐿⁄𝑑𝛼 is negative. With reference to Table 3.1 𝑑𝐶_𝐿⁄𝑑𝛼 = 3.122 > 0, thus galloping will not occur for the K12 design for Bjørnafjorden bridge.

Static divergence

Static divergence is a buckling type instability of the bridge girder occurring at the wind speed where the wind induced external moment acting on the girder exceeds the structural capacity. An estimate of the wind speed for onset for divergence 𝑉_𝑑𝑖𝑣 is given as /1/:

𝑉_𝑑𝑖𝑣 = 2𝜋𝑓_𝛼√ 2𝐼_𝛼𝑒𝑞

𝜌𝐵⁴𝑑𝐶_𝑀⁄𝑑𝛼 (4)

Where:

𝑓_𝛼 is the eigenfrequency of the lowest torsion mode having modal mass 𝑀_𝛼. 𝐼_𝛼𝑒𝑞 = ^𝑀^𝛼

∫ 𝜑(𝑥)₀^𝐿 ²𝑑𝑥 is the corresponding equivalent mass moment of inertia.

𝜌 = 1.25 kg/m³ is air density

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𝐵 = over-all girder width

The lowest torsion mode of the bridge is mode 11 displaying one partial half wave along the low level floating bridge, Figure 5.1, having an eigenfrequency 𝑓_𝛼 = 0.139 Hz and a modal mass 𝑀_𝛼 = 67.899∙10⁶ kg.

Figure 5.1 First torsion mode (mode 11) of Bjørnafjord K12 Alternative.

Inserting 𝑑𝐶_𝑀⁄𝑑𝛼 = 0.916 and the above structural properties in (4) a divergence wind speed 𝑉_𝑑𝑖𝑣 = 174.9 m/s which is well above the requirement of 81.7 m/s.

Classical flutter

Classical flutter involves as a minimum two modes of motion. A torsion mode and a vertical bending mode of similar mode shape but with a lower eigenfrequency. The critical wind speed for onset of classical flutter is reached when the wind loading on the bridge girder makes the bending and torsion frequencies equal thereby establishing a resonant exchange of energy between to two modes. This in turn leads to divergent coupled torsion bending oscillations of the bridge girder. In cases where more vertical modes exist below the torsion mode these vertical modes may couple to form a compound vertical mode shape which couples with the torsion mode at the onset of flutter.

Different methods exist for calculation of the flutter wind speed of a bridge deck. The present method outlined in section 6 is an expansion of the AMC method (Air material Command) which allows an arbitrary number of modes and degrees of freedom of a bridge deck to couple into flutter /6/. The input to the flutter calculation constitutes the modes assumed to couple into flutter, the corresponding modal masses and eigenfrequencies and aerodynamic derivatives particular to the bridge deck.

The present multi-mode flutter analysis of Bjørnafjorden K12 alternative assumes that the first torsion mode and 9 vertical bending modes with lower vibration frequencies than the selected torsional mode may couple into flutter, Figure 5.2. Eigenfrequencies and modal masses with of the modes are listed in Table 5.1.

Table 5.1 Lowest vertical and torsion modal masses and eigenfrequencies. Bjørnafjord K12 alternative.

Mode 15 Mode 16 Mode 17 Mode 18 Mode 19 Modal mass [kg] 16967 ∙10³ 18696 ∙10³ 28987∙10³ 71188 ∙10³ 43730 ∙10³

Eigenfrequency [Hz] 0.1449 0.1451 0.1543 01541 0.1536

Modal mass 57772 ∙10³ 52627 ∙10³ 56610∙10³ 91762 ∙10³ 73227 ∙10³

Eigenfrequency [Hz] 0.1543 0.1546 0.1548 0.1550 0.1608

0 1.033 10³ 2.066 10³ 3.099 10³ 4.132 10³ 5.165 10³

0.1 -

0.05 -

0 0.05 0.1

Length [m]

Modal displacement

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Figure 5.2 Flutter modes for Bjørnafjorden bridge K12 alternatives. Modes 15 – 25 (red) are vertical Mode 30 (blue) is torsion.

Mode 15

Mode 16

Mode 17

Mode 18

Mode 19

Mode 20

Mode 21

Mode 23

Mode 25

Mode 30

Concept development, floating bridge E39 Bjørnafjorden Appendix E – Enclosure 7

Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 7

10205546‐08‐NOT‐098 Bridge closure due to wind

MEMO

1 Bridge closure due to wind

2 References

Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 8

10205546‐08‐NOT‐176

Aerodynamic stability K11

MEMO

1 Wind load class

2 Critical wind speed for onset of aerodynamic instabilities

3 Bridge girder aerodynamic properties

4 Vortex induced vibrations

5 Verification of aerodynamic instabilities

6 Multimode flutter theory

7 References

Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 9

10205546-08-NOT-183

Inhomogeneity in wind – effects on K12

INTERNAL - MEMO

Table of Contents

1 Introduction

2 Implementation – calculation of displacement

3 Verification of calculation method.

4 Analysis and Results

5 Results and Discussion

6 References

7 Appendices

Concept development, floating bridge E39 Bjørnafjorden

Appendix E – Enclosure 10

10205546-08-NOT-184

Aerodynamic stability of K12

MEMO

Table of Contents

1 Wind load class

2 Critical wind speed for onset of aerodynamic instabilities

3 Bridge girder aerodynamic properties

4 Vortex induced vibrations

5 Verification of aerodynamic instabilities